Added-mass instability is known to be an important issue in the partitioned approach for fluid-structure interaction (FSI) solvers. Despite the implementation of the implicit approach, convergence of solution can be difficult to achieve. Relaxation may be applied to improve this implicitness of the partitioned algorithm, but this commonly leads to a significant increase in computational time. This is because the critical relaxation factor that allows stability of the coupling tends to be impractically small. In this study, a mathematical analysis for optimizing numerical performance based on different time integration schemes that pertain to both the fluid and solid accelerations is presented. The aim is to determine the most efficient configuration for the FSI architecture. Both theoretical and numerical results suggest that the choice of time integration schemes has a significant influence on the stability of FSI coupling. This concludes that, in addition to material and its geometric properties, the choice of time integration schemes is important in determining the stability of the numerical computation. A proper selection of the associated parameters can improve performance considerably by influencing the condition of coupling stability.